Maung and Paris [Internat J Intell Syst 1990, 5(5), 595–603] have shown that, in the general case, solving causal networks using maximum entropy techniques is NP complete. This paper considers multivalued causal inverted multiway trees, a nontrivial class of causal networks, in which any event can be influenced by any number of other events but itself only influences at most one event. We show that for this class of causal networks, maximum entropy can be used to find minimally prejudiced estimates for missing information. The techniques required for the current problem are substantially different from those used in the case of causal multiway trees in that nonlinear constraints arising from independence have to be incorporated. In addition, a new algebraic method is presented which isolates an unknown Lagrange multiplier by using the quotient of two pairs of state probabilities. Equating the joint probability distributions given by the Bayesian and maximum entropy models enables the Lagrange multipliers of the latter to be determined. An efficient iterative tree traversal algorithm which converges to the minimally prejudiced estimates for the missing information is described. When this information is added to that already provided, any existing method for updating the causal network can be used. ©1999 John Wiley & Sons, Inc.

The desire to use Causal Networks as Expert Systems even when the causal information is incomplete and/or when non-causal information is available has led researchers to look into the possibility of utilising Maximum Entropy. If this approach is taken, the known information is supplemented by maximising entropy to provide a unique initial probability distribution which would otherwise have been a consequence of the known information and the independence relationships implied by the network. Traditional maximising techniques can be used if the constraints are linear but the independence relationships give rise to non-linear constraints. This paper extends traditional maximising techniques to incorporate those types of non-linear constraints that arise from the independence relationships and presents an algorithm for implementing the extended method. Maximising entropy does not involve the concept of "causal" information. Consequently, the extended method will accept any mutually consistent set of conditional probabilities and expressions of independence. The paper provides a small example of how this property can be used to provide complete causal information, for use in a causal network, when the known information is incomplete and not in a causal form.

Expert systems that use causal probabilistic networks require the user to supply complete causal information regarding the causal probabilities to be used. This paper describes a method using the maximum entropy formalism that enables such expert systems to operate with incomplete causal information for certain classes of causal networks. It has been shown that, in the general case, solving causal networks using maximum entropy techniques is NP-complete. However, we show that for multivalued causal multiway trees—a nontrivial class of causal networks—the problem of estimating missing information is only linear. © 1998 John Wiley & Sons, Inc.